\begin{figure*}[!t]
\centering
        \subfigure[闭合位置：下线圈通电]{
            \label{d2} %% label for second subfigure
            \includegraphics[width=4cm]{down24}}
        \subfigure[断开位置：下线圈通电]{
            \label{d1} %% label for first subfigure
            \includegraphics[width=3.9cm]{down0}}       %\hspace{2mm}
        \subfigure[断开位置： 上线圈通电]{
            \label{u1} %% label for first subfigure
            \includegraphics[width=4cm]{up0}}
        %\hspace{2mm}
        \subfigure[闭合位置： 上线圈通电]{
            \label{u2} %% label for second subfigure
            \includegraphics[width=4.1cm]{up24}}
    \caption{制动器不同位置的磁势(Wb/m)分布
  }
    \label{mf} %% label for entire figure
\end{figure*}
\section{结果和讨论}\label{result}
%In this section, the results from the simulations and experiments are compared to verify the accuracy and effectiveness of the proposed method.
在这部分，我们通过对比了仿真的结果和试验结果来验证了所提出的的方法的准确性，并通过与其他方法进行对比，证明了该方法的计算效率。
\subsection{磁场}
%\begin{figure}[ht]
%    \centering
%        \subfigure[]{
%            \label{p1} %% label for first subfigure
%            \includegraphics[width=4cm]{pm0}}
%        %\hspace{2mm}
%        \subfigure[]{
%            \label{p2} %% label for second subfigure
%            \includegraphics[width=4cm]{pm24}}
%    \caption{.}
%    \label{mfpm} %% label for entire figure
%\end{figure}
%\begin{figure}[ht]
%    \centering
%        \subfigure[]{
%            \label{d1} %% label for first subfigure
%            \includegraphics[width=4cm]{down0}}
%        %\hspace{2mm}
%        \subfigure[]{
%            \label{d2} %% label for second subfigure
%            \includegraphics[width=4cm]{down24}}
%    \caption{.}
%    \label{mfdown} %% label for entire figure
%\end{figure}
%\begin{figure}[ht]
%    \centering
%        \subfigure[]{
%            \label{u1} %% label for first subfigure
%            \includegraphics[width=4cm]{up0}}
%        %\hspace{2mm}
%        \subfigure[]{
%            \label{u2} %% label for second subfigure
%            \includegraphics[width=4cm]{up24}}
%    \caption{.}
%    \label{mfup} %% label for entire figure
%\end{figure}
%Fig. \ref{mf} shows the distribution of magnetic potential in different positions of the actuator. It includes two kinds of operating positions: open and closed, and two modes of the coils. The coil working voltage is 28 V. From the pictures, we can further explore the variation of magnetic strength with respect to the changes in structure and current sources. In Fig. \ref{d2} and Fig. \ref{d1}, when the lower coil is working, the magnetic potential in the bottom region of actuator is increasing with the armature moving from closed position to open position. In Fig. \ref{u1} and Fig. \ref{u2}, when the upper coil is working, the magnetic potential in the top region of actuator is increasing with the armature moving from open position to closed position. In every new position, a closed magnetic loop is formed which makes the attraction more stable.
图 \ref{mf}是制动器衔铁在不同位置时的磁势的分布图。它包含两个位置：断开（open）和闭合（closed）。线圈的工作电压为28V。从图中，我们可以进一步探索磁场强度随着衔铁位置和电流密度发生变化的规律。在图 \ref{d2}和图 \ref{d1}，当下线圈工作时，随着衔铁不断从闭合位置移动到断开位置，触器底部区域的磁势不断增大。在图 \ref{u1}和图 \ref{u2}，当上线圈工作时，随着衔铁不断从断开位置移动到断闭合位置，触器顶部区域的磁势不断增大。在每一个新的位置，都会形成一个闭合磁路，使得接触器能够达到新的稳定状态。
\subsection{电磁力}
%The comparisons of magnetic force is illustrated in Fig. \ref{aaa}. The three curves show good accordance in the trends of force. It can be accepted that there are errors in some positions, especially at the closed and open position where magnetic field strength can vary fast. In Fig. \ref{pm}, the force in armature varies from negative to positive which means it has a balanced situation in the middle position; that is why this type of contactor has two steady states and saves energy.
如图 \ref{aaa}所示，为电磁力的数据对比。三条曲线表现出了很好的一致性。但是，在一些位置仍然有很大的误差，尤其是断开和导通的位置，可能原因是这部分的磁场变化幅度比较大。在图 \ref{pm}中，衔铁上的吸力由负变正，这意味着在运动的中间位置有一个平衡点，这也是为什么这种类型的接触器有两个稳定状态。

%Fig. \ref{28Vdown} illustrates the magnetic force when the bottom coil is on. With the help of coil, the armature moves from closed position to open position.
图 \ref{28Vdown}描述的是当下线圈上电时，电磁力的变化曲线。由于线圈提供的电磁力的帮助，衔铁从闭合位置运动到断开位置。

%On the contrary, Fig. \ref{28Vup} shows the force when the top coil is on and the armature moves from open position to closed position.
与之相反，图 \ref{28Vup}描述的是当上线圈上电时，电磁力的变化曲线。由于线圈提供的电磁力的帮助，衔铁从断开位置运动到闭合位置。
\begin{figure*}[t]
    \centering
        \subfigure[未通电时电磁力随衔铁位移变化曲线。 (衔铁从断开位置开始移动)]{
            \label{pm} %% label for second subfigure
            \includegraphics[width=5.7cm]{pm}}
        \subfigure[下线圈通电时电磁力随衔铁位移变化曲线。 (衔铁从断开位置开始移动)]{
            \label{28Vdown} %% label for first subfigure
            \includegraphics[width=5.5cm]{28Vdown}}       %\hspace{2mm}
        \subfigure[上线圈通电时电磁力随衔铁位移变化曲线。 (衔铁从断开位置开始移动)]{
            \label{28Vup} %% label for first subfigure
            \includegraphics[width=5.5cm]{28Vup}}
    \caption{不同工作条件下，电磁力随衔铁位移变化曲线
   }
    \label{aaa} %% label for entire figure
\end{figure*}
\subsection{计算耗时}
%Fig. \ref{tlm1} describes the comparison of computation time between traditional N-R method and TLM method proposed in this paper. The results are obtained on  a multi-core workstation which has two Intel Xeon Processor E5-2670 v3 (30M Cache, 2.30 GHz). Both the methods are implemented by using a parallel sparse matrix solver SuperLU\_MT \cite{superlu_ug99}.
图 \ref{tlm1}是分别使用牛顿迭代法和TLM方法进行计算后的计算时间的对比。程序运行在一个多核的工作站，配置为两个Intel Xeon Processor E5-2670 v3 (30M Cache, 2.30 GHz)处理器。两种方法都采用了并行的稀疏矩阵求解器SuperLU\_MT \cite{superlu_ug99}。
\begin{figure*}[t]
\centering
        \subfigure[N-R 迭代的单步计算时间]{
            \label{t1} %% label for first subfigure
            \includegraphics[width=8.5cm]{one}}
        %\hspace{2mm}
        \subfigure[TLM 迭代的单步计算时间]{
            \label{t2} %% label for second subfigure
            \includegraphics[width=8cm]{tlmsingle}}
        \subfigure[N-R 迭代的总共计算时间]{
            \label{n1} %% label for first subfigure
            \includegraphics[width=8cm]{nra}}
        %\hspace{2mm}
        \subfigure[TLM 迭代的总共计算时间]{
            \label{n2} %% label for second subfigure
            \includegraphics[width=8cm]{tlma}}
    \caption{随着CPU核心数和有限元分网大小变化两种方法运行时间对比
    }
    \label{tlm1} %% label for entire figure
\end{figure*}
%\begin{figure}[ht]
%
%    \caption{Comparison of total execution time with respect to CPU cores used and number of FEM elements
%    }
%    \label{nr1} %% label for entire figure
%\end{figure}

%The results in Fig. \ref{t1} and Fig. \ref{t2} show the computation time of one iteration step in N-R and TLM respectively as the FEM elements size increases and more computer cores are used. Although a parallel sparse matrix solver is used, the time in single step of N-R method still has a great gap and is more than 40 times larger than that of TLM. It can be understood that the LU factorization is dominant in one step of N-R. With the increase of cores, the time in single step of TLM reduces benefiting from the level schedule method in triangle solve. Both the methods obtain the largest parallel efficiency using approximately 8 CPU cores.
图 \ref{t1}和图 \ref{t2}分别是随着有限元分网单元数目的增加以及计算所有CPU核数的增加，牛顿迭代和TLM迭代单步所消耗的时间对比。尽管我们采用了一个并行的稀疏矩阵求解器，但是牛顿迭代法的单步计算时间与TLM相比，有很大的差距，比TLM慢40多倍。这是因为LU分解在矩阵的求解当中占据着主要的时间。随着计算核数的增加，通过采用级别调度法，TLM的单步计算时间取得了一定的并行收益。这两种方法都在8核的计算情况下获得了最大的并行效率。

%Fig. \ref{n1} and Fig. \ref{n2}  show the comparison of total computation time between N-R and TLM. And Table \ref{tabstep} lists the iteration steps of both methods with respect to different mesh size. Although the iteration steps in N-R is several times of TLM's, the TLM still occupies a dominant position in time consumption. The parallel strategies prove to be effective, especially when we increase the elements size, the parallel acceleration effect starts to be obvious.
图 \ref{n1}和图 \ref{n2}是牛顿迭代与TLM迭代的总共计算时间的对比。表 \ref{tabstep}是在不同分网大小下两种方法的迭代步数。尽管TLM所用的迭代步数是牛顿迭代的好几倍，但是TLM在时间消耗上仍然占有优势。并行计算方法被证明是有效的，尤其是当分网单元不断增多，并行效果变得更加明显。
\begin{table*}[t]
\caption{TLM与N-R迭代步数对比}\label{tabstep}
{\begin{tabular*}{20pc}{@{\extracolsep{\fill}}l|ll lll@{}}\toprule
分网大小 & 6690 & 10635 & 19221 & 26637 & 40318 \\

N-R 迭代步数 & 9 & 10 & 10 & 11 & 11 \\

TLM 迭代步数 & 65 & 69 & 73 & 81 & 90 \\
\bottomrule
\end{tabular*}}{}
\end{table*}
\subsection{讨论}
%Overall, this paper has described an optimised TLM iteration in parallel and its application in an axisymmetrical actuator with permanent magnet. Compared with previous works on TLM, we noticed that a pure resistance network is not enough to represent an FEM problem. Thus, we proposed a hybrid circuit network of resistances and voltage controlled current sources to model it. The most important feature of TLM iteration is that admittance matrix $Y$ remains unchanged during the iteration process. The improvement of speed is evident when the N-R and TLM programs run using the same number of CPU cores. Furthermore, we fully exploit these characteristics by adopting a level schedule method to solve the triangular matrices obtained from LU factorization. Results show that this method reduces computation time of the single step iteration as expected. %The results show the accuracy of the model and the effectiveness of computation schemes.
总的来说，这篇文章提出了一种优化的并行TLM迭代方法，并将它应用到了一种含永磁的轴对称制动器上。与之前的TLM方面的工作相比，我们注意到一个纯电阻的网络不足以来代表一个有限元问题。因此，我们提出了一个纯电阻和电压控制电流源的混合电路网络来进行等效建模。TLM迭代的最重要的特征是迭代过程当中传输线导纳矩阵$Y$保持不变。当牛顿迭代和TLM迭代使用相同的CPU核心来进行计算的时候，TLM的速度提升非常明显。为了更好的利用好TLM的这个特性，我们进一步地采用了一种级别调度法来并行地完成三角求解过程。结果表明这种方法极大地减少了单步迭代的计算时间。
%Unlike N-R iteration, the TLM is relatively easy to implement. However, we have to point out that the convergence speed of TLM is not as good as N-R's. Additional calculations must be done to improve it. Thus, we pay attention to the determination of the initial value of transmission line by using a N-R preconditioner. It is necessary and worthwhile to adopt a N-R preconditioner if we want to use TLM iteration better in practice. %In particular of large scale computation of mesh networks, the effect will be more obvious from our analysis of results.
与牛顿迭代法不同，TLM相对更加容易实现。但是，我们必须指出，TLM的收敛速度确实不如牛顿迭代快，必须通过额外的计算来提高TLM的收敛速度。因此，我们使用牛顿预处理器来提高其收敛速度。计算表明，如果我们想要使TLM方法有更好的实用价值，这是非常必要的步骤。
%Furthermore, we will try to improve our research and propose a practical scheme in 3-D magnetic field analysis using FEM to solve a series of problems in electromagnetic in the future.

